Gross and Siebert have recently proposed an "intrinsic" programme for studying mirror symmetry. In this talk, we will discuss a symplectic interpretation of some of their ideas in the setting of affine log Calabi-Yau varieties. Namely, we describe work in progress which shows that, under suitable assumptions, the wrapped Fukaya category of such a variety X gives an intrinsic "categorical crepant resolution" of Spec(SH0(X)). No background in mirror symmetry will be assumed for the talk.
Skew Calabi-Yau algebras are generalizations of Calabi-Yau algebras due to Reyes, Rogalski, and Zhang. Within the graded (associative and unital) algebras over a field k, they form the non-commutative analogues of the regular algebras. As a special feature, such an algebra A is equipped with its so-called Nakayama automorphism φ. The talk will present ongoing investigations on the presentations of these algebras by generators and relations taking into account their homological specificities. Such presentations are well-known for Calabi-Yau algebras (after Ginzburg, Bocklandt and van den Bergh) and also for Koszul skew Calabi-Yau algebras (after Bocklandt, Wemyss and Schedler). The general situation involves the interaction of the A∞-Yoneda algebra E(A) := ExtA(k,k) with the Nakayama automorphism φ, and also the A∞-Yoneda algebra E(A[x,φ]) of the Ore extension A[x,φ] of A by φ. More precisely, one is particularly intereseted in minimal models of these A∞-algebras. After having presented all these concepts, I will discuss the relationship between these minimal models as well as consequences in terms of presentations of A.
Consider a Calabi-Yau manifold which arises as a member of a Lefschetz pencil of anticanonical hypersurfaces in a Fano variety. The Fukaya categories of such manifolds have particularly nice properties. I will review this (partly still conjectural) picture, and how it constrains the field of definition of the Fukaya category.
Consider a Lagrangian torus fibration à la SYZ over a non-compact base. Using techniques from this arXiv paper, I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is generally not exact even near infinity. The construction allows the formulation of a version of the homological mirror symmetry conjecture for open manifolds which are not exact near infinity. According to time constraints, I will apply this to prove homological mirror symmetry in the case where the A-model is the complement of an anti-canonical divisor in a toric Calabi Yau manifold.
I will report on joint work with Nick Sheridan concerning structural aspects of mirror symmetry for Calabi-Yau manifolds. We show (i) that Kontsevich's homological mirror symmetry (HMS) conjecture is a consequence of a fragment of the same conjecture which we expect to be much more amenable to proof; and, in ongoing work, (ii) that from HMS one can deduce (some of) the expected equalities between genus-zero Gromov-Witten invariants of a CY manifold and the Yukawa couplings of its mirror.
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